FloraC
Joined 30 March 2020
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I explored and documented the [[sensamagic dominant chord]]. I explored the [[canou family]] of temperaments, and a few others in [[User:FloraC/Temperament proposal]]. | I explored and documented the [[sensamagic dominant chord]]. I explored the [[canou family]] of temperaments, and a few others in [[User:FloraC/Temperament proposal]]. | ||
Long term projects: | |||
* Cleanup for all temperament pages | |||
* Rework scale trees for mos pages | |||
* Work out RTT tables for all edos listed in the 13-limit page (relative error < 5.5% and cut off at 494) | |||
== Tools == | == Tools == | ||
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=== 3-limit TE tuning of ets === | === 3-limit TE tuning of ets === | ||
Given a val A, we have Tenney-weighted val V | {{Databox|Detail| | ||
Given a val A, we have Tenney-weighted val V = AW, where W is the Tenney-weighting matrix. | |||
If T is the Tenney-weighted tuning map, then for any et, for obvious reasons, | If T is the Tenney-weighted tuning map, then for any et, for obvious reasons, | ||
[math]t_2/v_2 | [math]t_2/v_2 = t_1/v_1[/math] | ||
Let ''c'' be the coefficient of TE-weighted tuning map ''c'' | Let ''c'' be the coefficient of TE-weighted tuning map ''c'' = ''t''<sub>2</sub>/''t''<sub>1</sub> = ''v''<sub>2</sub>/''v''<sub>1</sub> | ||
Let ''e'' be the [[TE error]] in Breed's RMS, and J be the [[JIP]], then | Let ''e'' be the [[TE error]] in Breed's RMS, and J be the [[JIP]], then | ||
[math]e | [math]e = {{!}}{{!}}T - J{{!}}{{!}}_\text {RMS} = \sqrt {\frac {(t_1 - 1)^2 + (t_2 - 1)^2)}{2} }[/math] | ||
Since | Since | ||
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[math] | [math] | ||
(t_1 - 1)^2 + (t_2 - 1)^2 \\ | (t_1 - 1)^2 + (t_2 - 1)^2 \\ | ||
= t_1^2 - 2t_1 + 1 + c^2 t_1^2 - 2c t_1 + 1 \\ | |||
= (c^2 + 1)t_1^2 - 2(c + 1)t_1 + 2 | |||
[/math] | [/math] | ||
has minimum at | has minimum at | ||
[math]t_1 | [math]t_1 = \frac{c + 1}{c^2 + 1} = \frac {v_1 (v_1 + v_2)}{v_1^2 + v_2^2}[/math] | ||
and ''f'' (''x'') | and ''f'' (''x'') = sqrt (''x''/2) is a monotonously increasing function | ||
''e'' has the same minimum point. | ''e'' has the same minimum point. | ||
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[math] | [math] | ||
t_i | t_i = \frac {v_i (v_1 + v_2)}{v_1^2 + v_2^2}, i = 1, 2 \\ | ||
e | e = \frac { {{!}}v_1 - v_2{{!}} }{\sqrt {2(v_1^2 + v_2^2)} } | ||
[/math] | [/math] | ||
}} | |||
=== 3-limit TOP tuning of ets === | === 3-limit TOP tuning of ets === | ||
{{Databox|Detail| | |||
This part is deduced from Paul Erlich's ''Middle Path''. | This part is deduced from Paul Erlich's ''Middle Path''. | ||
[math] | [math] | ||
t_i | t_i = \frac {2v_i}{v_1 + v_2}, i = 1, 2 \\ | ||
e | e = \frac { {{!}}v_1 - v_2{{!}} }{v_1 + v_2} | ||
[/math] | [/math] | ||
This ''e'' is also the amount to stretch or compress each prime. | This ''e'' is also the amount to stretch or compress each prime. | ||
}} | |||
=== General TE tuning of ets === | === General TE tuning of ets === | ||
{{Databox|Detail| | |||
[math]c_i | This time we have a sequence c = {''c''<sub>''n''</sub>}, where | ||
[math]c_i = v_i/v_1, i = 1, 2, \ldots, n[/math] | |||
And just proceed as before, | And just proceed as before, | ||
[math]t_1 | [math]t_1 = \frac {\sum \vec c}{\vec c^\mathsf T \vec c} = \frac {v_1 \sum V}{VV^\mathsf T}[/math] | ||
Substitute ''t''<sub>''i''</sub>/''c''<sub>''i''</sub> for ''t''<sub>1</sub>, | Substitute ''t''<sub>''i''</sub>/''c''<sub>''i''</sub> for ''t''<sub>1</sub>, | ||
[math] | [math] | ||
t_i | t_i = \frac {v_i \sum V}{VV^\mathsf T}, i = 1, 2, \ldots, n \\ | ||
e | e = \sqrt {1 - \frac {(\sum V)^2}{n VV^\mathsf T} } | ||
[/math] | [/math] | ||
}} | |||
=== Notes === | === Notes === | ||